Tim

Solved 50

• §16 — the rank function of the matroid $M'$Proved

  Oct 2026

• Lemma 10 — the support of a point of $H$ is a union of circuitsProved

  Oct 2026

• Lemma 11 — two points of $H$ supported on the same circuit are proportionalProved

  Oct 2026

• p. 533 — the matroid $M'$ is the matroid of a matrix of integers mod 2Proved

  Oct 2026

• §14 (14.1) — every real matrix has a circuit matrixProved

  Oct 2026

• §12 — the columns of a real matrix form a matroidProved

  Oct 2026

• §16 — the seven-element matroid $M'$ corresponds to no real matrixProved

  Oct 2026

• Theorem 27 — a subspace of $E_n$ has a unique associated matroidProved

  Oct 2026

• Theorem 22 — every matroid has a dualProved

  Oct 2026

• Theorem 21 — duality is symmetricProved

  Oct 2026

• Theorem 23 — duals iff bases correspond to base complementsProved

  Oct 2026

• Theorem 20 — a dual has rank $n(M)$ and nullity $r(M)$Proved

  Oct 2026

• Theorem 8 — an independent set extends to a base by elements of a given baseProved

  Oct 2026

• Theorem 7 — $B$ is a base iff $r(B)=r(M)$ and $n(B)=0$Proved

  Oct 2026

• Theorem 28 — orthogonal subspaces have dual associated matroidsProved

  Oct 2026

• Theorem 14 — distinct components are disjointProved

  Oct 2026

• Theorem 13 — two non-separable sets with a common element have a non-separable unionProved

  Oct 2026

• Theorem 12 — a non-separable set lies inside one part of a rank-additive unionProved

  Oct 2026

• Theorem 11 — rank additivity passes to subsets of the two partsProved

  Oct 2026

• Lemma 9 — a non-separable union of two disjoint parts has a circuit meeting bothProved

  Oct 2026

• Theorem 15 — a matroid is the sum of its components in a unique mannerProved

  Oct 2026

• Theorem 18 — three characterizations of the componentsProved

  Oct 2026

• Theorem 17 — building a non-separable matroid from a circuit by adding circuitsProved

  Oct 2026

• Theorem 16 — non-separable of nullity 1 iff circuitProved

  Oct 2026

• Theorem 19 — two elements share a component iff some circuit contains bothProved

  Oct 2026

• §4 — the independent sets of a rank system satisfy (I₂)Proved

  Oct 2026

• Lemma 4 — $\Delta(M + N, e) \le \Delta(M, e)$Proved

  Oct 2026

• Theorem 3 — submodularity of rank, (3.2) and (3.3)Proved

  Oct 2026

• Lemma 2 — any subset of an independent set is independentProved

  Oct 2026

• Lemma 3 — $\Delta(M + e_2, e_1) \le \Delta(M, e_1)$Proved

  Oct 2026

• Lemma 1 — rank and nullity are nonnegative and monotoneProved

  Oct 2026

• §6 — the rank postulates (R) and the independence postulates (I) are equivalentProved

  Oct 2026

• §8 — the rank defined from circuits satisfies (R₁), (R₂), (R₃)Proved

  Oct 2026

• Lemma 8 — the circuit rank of a set is independent of the ordering of its elementsProved

  Oct 2026

• Lemma 7 — interchanging the last two elements does not change the circuit rankProved

  Oct 2026

• §5 — the circuits of a rank system satisfy (C₁) and (C₂)Proved

  Oct 2026

• Theorem 5 — the nullity counts the steps at which adding an element closes a circuitProved

  Oct 2026

• Theorem 4 — a circuit in N + e contains e iff e is dependent on NProved

  Oct 2026

• Lemma 6 — if e is dependent on P₁ but on no proper subset of P₁, then P₁ + e is a circuitProved

  Oct 2026

• Lemma 5 — each element of a circuit is dependent on the rest of the circuitProved

  Oct 2026

• Theorem 4.15 -- M-convex sets correspond to integer submodular functionsProved

  Oct 2026

• Theorem 4.3 -- the four exchange axiom variants are equivalentProved

  Oct 2026

• §8 — the rank postulates (R) and the circuit postulates (C) are equivalentProved

  Oct 2026

• Theorem 4.17 -- Frank's discrete separation theoremProved

  Oct 2026

• Theorem (P), p. 126 — the vertices of the polyhedron C are exactly the matching vectors of GProved

  Oct 2026

• Theorem 4.18 -- Edmonds's intersection theoremProved

  Oct 2026

• Theorem 1 (Minimal cut theorem), p. 400 — the maximal flow value equals the minimum value of a disconnecting setProved

  Oct 2026

• Theorem 1 — a matching is maximum iff no alternating chain connects two neutral pointsProved

  Oct 2026

• Berge (1957), Theorem 1, converse: a non-maximum matching has an alternating chain between two distinct neutral pointsProved

  Oct 2026

• Lemma 5 — at most one neutral point forces a maximum matchingProved

  Oct 2026

Posted 1

• Berge (1957), Theorem 1, converse: a non-maximum matching has an alternating chain between two distinct neutral pointsProved

  Oct 2026